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3

Find the points of discontinuity, if any, of the following functions :

Basic Idea:

A real function f is said to be continuous at x = c, where c is any point in the domain of f if :

where h is a very small ‘+ve’ no.

i.e. left hand limit as x → c (LHL) = right hand limit as x → c (RHL) = value of function at x = c.

This is very precise, using our fundamental idea of limit from class 11 we can summarise it as, A function is continuous at x = c if :

Here we have,

…Equation 1

Function is defined for all real numbers so we need to comment about its continuity for all numbers in its domain ( domain = set of numbers for which f is defined )

Let c is any random number such that c ≠ 0 [thus c being random number, it is able to include all numbers except 0 ]

f(c) = [ using eqn 1]

Clearly,

∴ We can say that f(x) is continuous for all x ≠ 0

As x = 0 is a point at which function is changing its nature so we need to check the continuity here.

Since, f(0) = 7 [using eqn 1]

NOTE : Idea of logarithmic limit and exponential limit –

You must have read such limits in class 11. You can verify these by expanding log(1+x) and e^{x} in its taylor form.

Numerator and denominator conditions also hold for this limit like sandwich theorem.

E.g :

But,

and,

= [Using logarithmic and exponential limit as explained above, we have:]

=

Thus,

∴ f(x) is discontinuous at x = 0

Hence, f is continuous for all x ≠ 0 but discontinuous at x = 0